is a ring homomorphism mapping to and V(I) is a Zariski-closed subset of Spec(R). To show that f* is continuous, you need to show that is a Zariski-closed subset in Spec(S) for an arbitrary Zariski closed subset V(I) in Spec(R). Verify that is (See here). Thus f* is continuous.
FYI, you can find some explanations about this problem in Dummit P.734 (proposition 55).